Calculus Multivariable, Second Edition
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Blank and Krantz's Calculus 2e brings together time-tested methods and innovative thinking to address the needs of today's students, who come from a wide range of backgrounds and look ahead to a variety of futures. Using meaningful examples, credible applications, and incisive technology, Blank and Krantz's Calculus 2e strives to empower students, enhance their critical thinking skills, and equip them with the knowledge and skills to succeed in the major or discipline they ultimately choose to study. Blank and Krantz’s engaging style and clear writing make the language of mathematics accessible, understandable and enjoyable, while maintaining high standards for mathematical rigor.

Blank and Krantz's Calculus 2e is available with WileyPLUS, an online teaching and learning environment initially developed for Calculus and Differential Equations courses. WileyPLUS integrates the complete digital textbook with powerful student and instructor resources as well as online auto-graded homework.


Brian E Blank is an associate professor of mathematics at Washington University in St. Louis. He received his Ph.D. in 1980 at Cornell University, with Anthony Knapp as advisor. His 20th century work involved harmonic analysis.

Steven G Krantz is an American scholar, mathematician, and writer. He has authored more than 235 research papers and more than 125 books. Additionally, Krantz has edited journals such as the Notices of the American Mathematical Society and The Journal of Geometric Analysis.



Supplementary Resources.


About the Authors.

C H A P T E R 9 Vectors.


1 Vectors in the Plane.

2 Vectors in Three-Dimensional Space.

3 The Dot Product and Applications.

4 The Cross Product and Triple Product.

5 Lines and Planes in Space.

Summary of Key Topics.

Review Exercises.

Genesis & Development 9.

C H A P T E R 1 0 Vector-Valued Functions.


1 Vector-Valued Functions—Limits, Derivatives, and Continuity.

2 Velocity and Acceleration.

3 Tangent Vectors and Arc Length.

4 Curvature.

5 Applications of Vector-Valued Functions to Motion.

Summary of Key Topics.

Review Exercises.

Genesis & Development 10.

C H A P T E R 1 1 Functions of Several Variables.


1 Functions of Several Variables.

2 Cylinders and Quadric Surfaces.

3 Limits and Continuity.

4 Partial Derivatives.

5 Differentiability and the Chain Rule.

6 Gradients and Directional Derivatives.

7 Tangent Planes.

8 Maximum-Minimum Problems.

9 Lagrange Multipliers.

Summary of Key Topics.

Review Exercises.

Genesis & Development 11.

C H A P T E R 1 2 Multiple Integrals.


1 Double Integrals Over Rectangular Regions.

2 Integration Over More General Regions.

3 Calculation of Volumes of Solids.

4 Polar Coordinates.

5 Integrating in Polar Coordinates.

6 Triple Integrals.

7 Physical Applications.

8 Other Coordinate Systems.

Summary of Key Topics.

Review Exercises.

Genesis & Development 12.

C H A P T E R 1 3 Vector Calculus.


1 Vector Fields.

2 Line Integrals.

3 Conservative Vector Fields and Path Independence.

4 Divergence, Gradient, and Curl.

5 Green’s Theorem.

6 Surface Integrals.

7 Stokes’s Theorem.

8 The Divergence Theorem.

Summary of Key Topics.

Review Exercises.

Genesis & Development 13.

Table of Integrals.

Formulas from Calculus: Single Variable.

Answers to Selected Exercises.